⟨a, b, c | ab=1, cbbcc=b⟩

Up:

Monoid enumeration

#7103 ⟨a, b, c | ab=1, cbacc=c⟩

#7106 ⟨a, b, c | ab=1, cbbcc=c⟩

Properties

Sum of relation sides is 8

Infinite non-cancellative non-commutative monoid

Element a has infinite order

Not left cancellative, because left multiplication by a is not injective:

a ⋅ cec³a = a and a ⋅ 1 = a, however cec³a ≠ 1

Not right cancellative, because right multiplication by cec³ is not injective:

cec³a ⋅ cec³ = cec³ and 1 ⋅ cec³ = cec³, however cec³a ≠ 1

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Auxiliary generators:
1. d = cb
2. e = bdbc
Reduction order:
- Left-to-right recursive path with deg(c) = deg(d) = 0, c < d; deg(a) = deg(e) = deg(b) = 1, a < e < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	ced ⇒ edcdc²	[29]
2.	cecd ⇒ edc²	[25]
3.	cec²d ⇒ ec	[19]
4.	de ⇒ e(dc)²	[28]
5.	c²ec³ ⇒ d	[21]
6.	dcec² ⇒ ce	[24]
7.	b ⇒ cec³	[18]
8.	ae ⇒ cec²	[23]
9.	acec³ ⇒ 1	[20]
10.	ce² ⇒ e(dc)³c²ec²	[31]
11.	(ce)² ⇒ edcdc³ec²	[30]
12.	cec²e ⇒ edc³ec²	[26]
13.	cec³e ⇒ (ec²)²	[17]
14.	cec⁴ec² ⇒ e	[22]

# abc:ab=1,cbbcc=b cd/aeb cb=d,bdbc=e morph:2/1,4/1 ced=edcdcc cecd=edcc ceccd=ec de=edcdc cceccc=d dcecc=ce b=ceccc ae=cecc aceccc=1 cee=edcdcdcccecc cece=edcdcccecc cecce=edcccecc ceccce=eccecc ceccccecc=e

#7105 ⟨a, b, c | ab=1, cbbcc=b⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#7103 ⟨a, b, c \| ab=1, cbacc=c⟩
Next:	#7106 ⟨a, b, c \| ab=1, cbbcc=c⟩